7 ALGEBRAIC PROPERTIES OF CONVOLUTION 277
(14.7.3) If the sequence (u^..., un) is convolvable, then so is the sequence
(184.108.40.206) (//! * fi2 * • • • * uny = £in * #„_! * • • • * /V
For (14.1.4 and 13.7.10) we have
if and only if
and these two integrals are equal.
On the other hand, if the sequence (A, u) is convolvable, it does not
necessarily follow that (/*, A) is convolvable (Problem 2). But if G is com-
mutative this will be the case, and we shall have A * \JL = fj. * A.
In particular, it follows from the preceding results that
(14.7.4) On the set M£(G) of bounded measures on G, the law of composition
(A, p) i-> A * IJL (together with the vector space structure) defines a C-algebra
structure\ the unit element is the Dirac measure se at the neutral element e ofG.
The set M£(G) of compactly supported measures on G is a subalgebra of
Mc(G). The algebra M£(G) is commutative if and only ifG is commutative.
The fact that G is commutative if M£(G) is commutative follows from the
If G is discrete, the algebra M£(G) consists of all linear combinations
]T as8&, where as = 0 for all but a finite number of points s e G (3.16.3), and
the formula (220.127.116.11) shows that
seG / \seG
This is what is called in algebra the group algebra of the group G over the
field C.f*4.(G), hence is (A * ^)-integrable, and we have